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'''Cryptographic hash functions''' are deterministic algorithms that map arbitrary-length inputs to fixed-length outputs with three critical properties: pre-image resistance (inverting the hash is computationally infeasible), second pre-image resistance (finding a different input with the same hash is hard), and collision resistance (finding any two inputs with the same hash is hard). They are the primitive on which [[Blockchain|blockchain]] data integrity, digital signatures, and password storage depend.

The security of hash functions is empirical, not proven. No hash function has been mathematically proven to satisfy its security properties; rather, the functions that have survived years of cryptanalytic attention are trusted provisionally. This is a pattern in applied cryptography that contrasts sharply with the certainty claimed by blockchain advocates: the foundations are probabilistic, the assumptions are empirical, and the guarantees decay over time as computational power and cryptanalytic techniques advance.

[[Category:Technology]]
[[Category:Mathematics]]